Question

A storage shed is to be built in the shape of a box with a square base. It is to have a volume of 686 cubic feet. The concrete for the base costs ​$5 per square​ foot, the material for the roof costs ​$9 per square​ foot, and the material for the sides costs ​$3.50 per square foot. Find the dimensions of the most economical shed.

Answer 1

Answer:

Therefore the dimension of box is 7 ft by 7 ft by 14 ft.

Step-by-step explanation:

Given that, a storage shade is be built in the shape of a box with a square base.

Let the height of the box be h and the length of one side of the square base be x.

The area of the square base is = side²

                                                    =x²

The volume of the box is = area of the base × height

                                          =x²h

According to the problem,

x²h=686

$\Rightarrow h=\frac{686}{x^2}$ .......(1)

The concrete for the base costs $5 per square foot.

The material for the base costs =$ 5x²

The material for the roof costs $9 per square foot.

The material cost for roof is =$9x²

The material for the sides costs $3.50  per square foot.

The material cost for sides =$(3.50× 4xh )

                                             =$14xh

Total cost =$(5x²+9x²+14xh)

                =$(14x²+14 xh)

Let

C = 14x²+14 xh

Putting $h=\frac{686}{x^2}$

$C=14x^2+14 x.\frac{686}{x^2}$

$\Rightarrow C=14x^2+\frac{9604}{x}$

Differentiating with respect to x

$C'= 28x-\frac{9604}{x^2}$

Again differentiating with respect to x

$C''= 28+\frac{19208}{x^2}$

To find the dimension set C'=0

$28x-\frac{9604}{x^2}=0$

$\Rightarrow 28x=\frac{9604}{x^2}$

$\Rightarrow x^3=\frac{9604}{28}$

$\Rightarrow x=7$

Now,

$C''|_{x=7}= 28+\frac{19208}{7^2}>0$

Since at x=7, C''>0, So at x=7 , The cost of material will be minimum.

The height of the box is  $h=\frac{686}{x^2}$

                                            $=\frac{686}{7^2}$

                                            =14 foot

Therefore the dimension of box is 7 ft by 7 ft by 14 ft.

The cost of the material is $=14x^2+\frac{9604}{x}$

                                              $=14(7)^2+\frac{9604}{7}$

                                             =$2,058